O(n)
O(log n)
h
k
L
f
X f(X)
T f X
Y
m
f
˜
X = X
k = 1 . . . , m
f
(k)
= L(
˜
mX)
f f
(k)
f
˜
mX f
(k)
(
˜
mX)
f T (f, X, Y )
f
400
1 4
y
x P (x)
P (y|x) P (x)
P (x) P (y | x)
y x y x
y x
P
1
P
2
P
1
P
2
P
1
P
2
selection switch
x
(1)
x
(2)
x
(3)
h
(3)
h
(2)
h
(1)
y
y x
x
(1)
x
(2)
x
(3)
P
1
P
2
P
1
P
2
(x, y) x
y
x
y
P (x) (x, y)
P (y | x)
X
x
1
x
2
P (x) P (x, y) P (y | x)
log P (y | x)
log P (x) log P (x, y)
P (x) P (y | x)
P (x) P (y | x)
P (x)
x
y x
x y
P (y | x) x P (x)
P (x) P (y | x)
0 5 10 15 20
x
0.0
0.1
0.2
0.3
0.4
0.5
p(x)
y=1 y=2 y=3
Mixture model
x
y
p(x) y
P (x) E[y | x]
x P (y | x)
x
y
P (x)
P (y | x) P (y | x) P(x)
y x
P (x) P (y | x)
y x h
h x
P (h, x) = P (x | h)P (h).
P (x) =
P (x | h)p(h)dh
P (x) =
h
P (x | h)P (h).
x
h
x
y
y
y x
P (y | x) =
P (x | y)P (y)
P (x)
.
P (x) P (y | x)
h y = h
i
h
j
y h
x y P (x | y)
P (y)
P (x | y) P (y)
h P (x | h)
n
2
n
n
n
n
n
n
k
k
n
w
1
w
2
k n n
O(n
k
)
u v
f f(u) f(v)
(x, y) f(x) y
ˆ
f
n
n
n R
d
d
j=0
n
j
= O(n
d
),
d
f 2
d
O(2
d
)
2
d
n
k
k 1
d
L n
O
n
d
d(L1)
n
d
,
L k
O
k
(L1)+d
.
y = h
i
x
f x y
f(x) f(y)
x
P (x)
P (y | x) P (y | x)
P (x)
P (x) P (y | x)
y
i
x
f
i
(x) x
y
i
h
P (y
i
| x) P (h | x)
P (x | = i) i
x
x
P (h) =
i
P (h
i
)